We are well aware of anti-evolutionists' fondness for presenting their audiences with numbers of dizzying magnitude that they use to represent incredibly low probabilities for such events as the chance formation of a protein molecule, the origin of life, and the like. Thus they argue that it is irrational to believe that the event in question could have happened naturally (they mean "by chance") without the aid of intelligent design. In some cases, such as the chance formation of habitable planets, one may avoid a technical discussion of the physical processes involved and respond simply by pointing out that the universe is a very big place, containing countless galaxies, stars, and planetary systems, thus providing so much opportunity for the natural occurrence of the event in question that the probability may be quite high that such an event would occur somewhere. Furthermore, if the universe is infinite, providing the event with infinitely many chances to occur, then the occurrence of the event is a virtual certainty. Thus creationist probability arguments can often be undermined by pointing out that any event with a probability greater than 0, no matter how low, will be likely to happen if given enough opportunity, and sure to happen if opportunity is unlimited.

This principle is sometimes illustrated with the following thought experiment (of which the reader has probably heard one version or another): Suppose that a monkey, trained to hit the keys of a typewriter one by one in a truly random fashion, types forever, producing infinitely many pages of text. No one doubts that the monkey would type page after page of gibberish, but it follows from the above principle that sooner or later the monkey would type all of the works of Shakespeare from beginning to end, without error, solely by accident.

Unfortunately, this result of the thought experiment, and thus the principle itself, is sometimes explicitly rejected by creationists. One way of trying to justify their denial of this principle is by an appeal to what creationists refer to as Borel's single law of chance - a claim made by the French probability theorist Emile Borel. According to creationists, Borel's single law of chance says that any event with a probability lower than 1 in 1050 is so improbable as to be impossible (Kennedy 1980: 57; Ankerberg and Weldon 1998: 183; Harber 1998: 33; Mastropaolo 1999: iii). The implication is that, since the origin of life, the evolution of humans, and many other events may have a probability below this limit, they could not possibly have happened by chance no matter how much opportunity there may have been for them to occur.

Thus creationists attempt to protect their probability arguments from our sufficient opportunity principle by invoking this single beloved mathematical law. Borel did in fact propose such a law. However, just as creationists have misrepresented the second law of thermodynamics, so have they misrepresented Borel's law of chance. So what did Borel really mean? Here is an illustration.

Lightning Strikes - Often!

Hardly any of us really worries about getting struck by lightning. The probability that any individual will ever be struck by lightning is extremely low. But with so many people in the world, there is ample opportunity for this rare event to happen from time to time. It would be amazing if it never happened; and indeed many of us do know of such an event. Thus there are some highly improbable events that may be rationally expected to happen occasionally.

On the other hand, we can imagine other events (such as a monkey's accidentally typing Shakespeare) that are so improbable that the entire observable universe cannot provide enough opportunity for us rationally to expect the event in question to occur. Any event of this sort that has any probability at all is still possible - it is just that it would be foolish to bet on its occurrence, not only at a particular place or time, but anywhere ever (within the spatial and temporal confines of the observable universe). Borel said that such events, having a probability of no more than roughly 1 in 1050, never occur (Borel 1965: 57). But this law of chance is not literally true, for, as we shall see, such events can and do happen. I think that a more accurate way to say what Borel had in mind is that in reality, no such event can be rationally predicted ever to occur.

Unfortunately, because, I suspect, of the carelessness of creationists' research, they have failed to grasp Borel's law and instead have taken his claim at face value - as saying literally that events of such low probabilities cannot possibly occur! For example, according to Scott Huse, "[M]athematicians generally consider any event with a probability of less than 1 chance in 1050 as having a zero probability ([that is] it is impossible)" (Huse 1997: 123). So in effect we are told that according to Borel's single law of chance, even if the observable universe did provide unlimited opportunity for their occurrence, such events are just too improbable ever to occur (Ankerberg and Weldon 1998: 329-30). It is this claim with which I take issue (as would Borel), for though one need not be learned in mathematics to find the claim questionable, many laypeople, I fear, may find it all too easy to believe.

All Nonzero Probabilities Are Possible

The probability of an event is expressed as a real number from 0 to 1; the more probable the event, the higher the number. An event can have only one probability at any time, just as a person at any given time can have only one age. However, anti-evolutionists misconstrue Borel's law of chance to imply the absurdity that low-probability events are assigned 2 different probabilities - their true probability and a probability of 0.

By way of example, suppose that one were to program a computer to generate 100 random digits. There would be 10100 equally likely possible outcomes. The probability of any given outcome would thus be 10-100. Applying the creationist "law of chance", we would have to conclude that any conceivable outcome, because it has a probability less than 1 in 1050, is literally impossible, having no chance of occurring and thus having a probability of 0 (see the Huse quote above). But clearly no event can have a probability of 1 in 10100and a probability of 0 (unless we think that 1/10100 = 0, which is as false as the claim that 2 + 2 = 5). Moreover, since the conceivable outcomes are what mathematicians call mutually exclusive and jointly exhaustive, the sum of all their individual probabilities must equal 1, which they cannot do if they are all 0.

Fortunately, one need only carry out this experiment to see the anti-evolutionists' version of this "law of chance" falsified. For surely some outcome must be realized when we instruct the computer to select 100 random digits, despite the fact that the calculated probability of each outcome that the computer could produce falls far below the supposed threshold of possibility. (Borel, on the other hand, would say that no preconceived outcome could be rationally expected to occur, because the probability of successfully guessing the outcome in advance is too low for it to be expected to happen in the real world.) Thus we see that the anti-evolutionist appeal to Borel's law of chance fails to refute the principle that any event with a positive probability, no matter how small, is bound to happen somewhere sometime if given infinitely many chances.

Typing Monkeys and the Classics

Another way that anti-evolutionists try to get around this principle is simply by a dubious appeal to common sense. As one apologist argues, "It does not matter how much time we give nature; the large numerical odds simply are irrelevant: we must simply admit that no matter how much time and how much luck, evolution could not have happened" (Lutzer 1998: 159). Unfortunately, common sense is not always, and certainly not in this case, a reliable guide to mathematical truth.

For example, Patrick Glynn, in God: The Evidence, criticizes our thought experiment about the endlessly typing monkey in this way:

[I]t does not matter if there is an infinity of days. ... It is a gross fallacy to suppose that the quantity of days or time available changes anything. (To put the proposition mathematically, the probability on any given day that the monkey will type the works of Shakespeare . . . is not one in some very, very large number; it is zero.) Randomness does not engender order on any appreciable scale, no matter how many billions of years or opportunities you give it (Glynn 1997: 46).

But instead of relying on gut instinct, let us see if a far more reliable appeal to probability theory cannot shed some light on the subject. (Borel's law of chance is of no use to us here, for it is applicable only to real world cases, not hypothetical cases like this where we have eternity at our disposal.)

Let us say that Shakespeare's Hamlet is x typed pages long, y is the number of characters that can fit on a page, and z is the number of characters on a typewriter. Thus a text x pages long contains xy characters, each of which could be any one of z possibilities. There are then zxy possible ways of randomly typing x pages of text, all of which are equally likely and exactly one of which is Hamlet. Now suppose that we divide the monkey's work into trials, the first trial consisting of the first x pages typed, the second trial consisting of the second x pages, and so on. Since the monkey will ultimately type infinitely many pages, he will ultimately type infinitely many trials. Each trial is an opportunity for the monkey to type Hamlet. (I am, of course, ignoring the possibility that the monkey might begin typing Hamlet midway through one trial and finish it midway through the next.)

On any given trial, the probability that the monkey will type Hamlet is 1/zxy, which we shall call p. And on any given trial, the probability that the monkey will fail to type Hamlet is 1-p, which we shall call q. Thus the probability of failure is q for the first trial, q for the second trial, q for the third trial, and so on. Consider now the probability of failure on the first 2 trials, which is q2, and the probability of failure on the first 3 trials, which is q3, and so on. We thus see that the probability of failure on all of the first n trials is qn. What then is the probability that the monkey will fail on all the trials, that he will never type Hamlet? Since there are infinitely many trials, the probability could be expressed as q raised to the power of infinity.

But what are we to make of this? Since zxy is a finite positive number, 1/zxy (the probability on any trial that the monkey will type Hamlet) must be greater than 0. And p = 1/zxy, so p > 0. Since p > 0, we know that 1-p is smaller than 1. And since 1-p = q, it follows that q must be smaller than 1. Thus we see that q, the probability on any given trial that the monkey will fail to type Hamlet, must be a real number greater than 0 and less than 1. Now if we pick any number on the number line greater than 0 and less than 1 (q is such a number) and multiply it by itself many times, thus raising it to higher and higher powers, the product will approach 0; the higher the power, the closer to 0 the product will be. If the power is infinite, the ultimate result, in the end, is 0. Therefore, q to the power of infinity is 0.

Now recall that q to the power of infinity is the probability that the monkey would never type Hamlet. And we have just seen that this probability is 0. This means that there is a probability of 1, or 100%, that the monkey will type Hamlet at least once over the course of eternity. It would be a miracle if he did not! The same goes for every other work of Shakespeare, as well as the Bible, War and Peace, this article, your personal diary, anything imaginable (of finite length) - you name it, the monkey will eventually type it.

In fact, this principle is not limited to this thought experiment about the monkey typing, but is applicable to any improbable event whatsoever. Provided it has a constant positive probability (to be represented by p), the event in question is certain to happen if given unlimited opportunity. Any attempt to deny this, whether based on Borel's law of chance, common horse sense, or anything else, is misguided.

The Heart of the Matter

Anti-evolutionists, of course, will continue to employ their probability arguments against the natural formation of proteins, cells, and the like, despite everything said in this article. There are two reasons for this. First, in all fairness, their probability arguments often cannot be adequately refuted without a highly technical scientific explanation of the physical processes involved in the "improbable" event in question, and no such discussion was attempted here for the same reason that none is often attempted in public discussions of the issues.

Second, and more importantly, even if all the scientific matters had been discussed, it would make no difference. The opponents of evolution are not interested in good science, and as I have attempted to show in this article, neither are they interested in good mathematics. Hence their arguments are not based on a complete and contemporary understanding of the scientific and mathematical principles that are relevant to the issue. This is yet another reason why creationist material has no business being taught in science classes - it threatens our students' education not only with bad science, but also bad mathematics.